Lixin X. Ding

An Orthogonal Multi-objective Evolutionary Algorithm for Multi-objective Optimization Problems with Constraints

In this paper, an orthogonal multi-objective evolutionary algorithm (OMOEA) is proposed for multi-objective optimization problems (MOPs) with constraints. Firstly, these constraints are taken into account when determining Pareto dominance. As a result, a strict partial-ordered relation is obtained, and feasibility is not considered later in the selection process. Then, the orthogonal design and the statistical optimal method are generalized to MOPs, and a new type of multi-objective evolutionary algorithm (MOEA) is constructed. In this framework, an original niche evolves first, and splits into a group of sub-niches. Then every sub-niche repeats the above process. Due to the uniformity of the search, the optimality of the statistics, and the exponential increase of the splitting frequency of the niches, OMOEA uses a deterministic search without blindness or stochasticity. It can soon yield a large set of solutions which converges to the Pareto-optimal set with high precision and uniform distribution. We take six test problems designed by Deb, Zitzler et al., and an engineering problem (W) with constraints provided by Ray et al. to test the new technique. The numerical experiments show that our algorithm is superior to other MOGAS and MOEAs, such as FFGA, NSGAII, SPEA2, and so on, in terms of the precision, quantity and distribution of solutions. Notably, for the engineering problem W, it finds the Pareto-optimal set, which was previously unknown.

An Efficient Multi-Objective Evolutionary Algorithm for Combinational Circuit Design

In this paper we introduce an efficient multi-objective evolutionary algorithm (EMOEA) to design circuits. The algorithm is based on non-dominated set for keeping diversity of the population and therefore, avoids trapping in local optimal. Encoding of the chromosome is based on J. F. Millers implementation, but we use efficient methods to evaluate and evolve circuits for speeding up the convergence of the algorithm. This algorithm evolves complex combinational circuits (such as 3-bit multiplier and 4 bit full adder) without too much long time evolution (commonly less than 5,000,000)

A lower-dimensional-search evolutionary algorithm and its application in constrained optimization problems

This paper proposes a new evolutionary algorithm, called lower-dimensional-search evolutionary algorithm (LDSEA). The crossover operator of the new algorithm searches a lower-dimensional neighbor of the parent points where the neighbor center is the barycenter of the parents therefore the new algorithm converges fast, especially for high-dimensional constrained optimization problems. The niche-impaction operator and the mutation operator preserve the diversity of the population to make the LDSEA algorithm not to be trapped in local optima as much as possible. Whats more is that the LDSEA algorithm is simple and easy to be implemented. We have used the 24 constrained benchmark problems [18] to test the LDSEA algorithm. The experimental results show it works better than or competitive to a known effective algorithm [7] for higher-dimensional constrained optimization problems.

Some theoretical results about the computation time of evolutionary algorithms

This paper focuses on the computation time of evolutionary algorithms. First, some exact expressions of the mean first hitting times of general evolutionary algorithms in finite search spaces are obtained theoretically by using the properties of Markov chain. Then, by introducing drift analysis and applying Dynkins Formula, the general upper and lower bounds of the mean first hitting times of evolutionary algorithms are given rigorously under some mild conditions. These results obtained in this paper, and the analytic methods used in this paper, are widely valid for analyzing the computation time of evolutionary algorithms in any search space(finite or infinite)as long as some simple technique processes are introduced.

Constrained optimization by the evolutionary algorithm with lower dimensional crossover and gradient-based mutation

This paper proposes a new evolutionary algorithm with lower dimensional crossover and gradient-based mutation for real-valued optimization problems with constraints. The crossover operator of the new algorithm searches a lower dimensional neighbor of the parent points where the neighbor center is the barycenter of the parents, and therefore the new algorithm converges fast. The gradient-based mutation is used to converge fast for the problems with equality constraints and active inequality constraints. And the new algorithm is simple and easy to be implemented. We have used 24 constrained benchmark problems to test the new algorithm. The experimental results show it works better than or competitive to a known effective algorithm.

An evolutionary algorithm of contracting search space based on partial ordering relation for constrained optimization problems

A new evolutionary algorithm, which can contract search space based on the partial ordering relation and is designed to solve nonlinear programming (NLP), is proposed in this paper. Firstly, the partial ordering relation is used for evaluating an individual, which ensures that individual competition is more impartial. Secondly, by taking advantage of incomplete evolution, which provides good individuals in short time, we can locate regions of optimal solutions and contract the search space and thus reduce the search space and increase the convergence rate. Thirdly, we prove that the algorithm can find optimal solutions. Finally, the algorithm can be easily parallelized. Numerical experiments demonstrate that our techniques are superior to other methods in terms of solution quality and robustness.

A new technique for assessing the diversity of close-Pareto-optimal front

The quality of an approximation set usually includes two aspects-- approaching distance and spreading diversity. This paper introduces a new technique for assessing the diversity of an approximation to an exact Pareto-optimal front. This diversity is assessed by using an ldquoexposure degreerdquo of the exact Pareto-optimal front against the approximation set. This new technique has three advantages: Firstly, The ldquoexposure degreerdquo combines the uniformity and the width of the spread into a direct physical sense. Secondly, it makes the approaching distance independent from the spreading diversity at the most. Thirdly, the new technique works well for problems with any number of objectives, while the widely used diversity metric proposed by Deb would work poor in problems with 3 objectives or over. Experimental computational results show that the new technique assesses the diversity well.

About the limit behaviors of the transition operators associated with EAs

This paper focuses on the limit behaviors of evolutionary algorithms based on finite search space by using the properties of Markov chains and Perron-Frobenius Theorem. Some convergence results of general square matrices are given, and some useful properties of homogeneous Markov chains with finite states are investigated. The geometric convergence rates of the transition operators, which is determined by the revised spectral of the corresponding transition matrix of a Markov chain associated with the EA considered here, are estimated. Some applications of the theoretical results in this paper are also discussed.

Both robust computation and mutation operation in dynamic evolutionary algorithm are based on orthogonal design

A robust dynamic evolutionary algorithm (labeled RODEA), where both the robust calculation and mutation operator are based on an orthogonal design, is proposed in this paper. Previous techniques calculate the mean effective objective (for robust) by using samples without much evenly distributing over the neighborhood. The samples by using orthogonal array distribute evenly. Therefore the calculation of mean effective objective more robust. The new technique is generalized from the ODEA algorithm [1]. An orthogonal design method is employed on the niches for the mutation operator to find a potentially good solution that may become the representative in the niche. The fitness of the offspring is therefore likely to be higher than that of its parent. We propose a complex benchmark, consisting of moving function peaks, to test our new approach. Numerical experiments show that the moving solutions of the algorithm are a little worse in objective value but robust.

An orthogonal dynamic evolutionary algorithm with niches

A new dynamic evolutionary algorithm based on orthogonal design (denoted by ODEA) is proposed in present paper. Its population does not consist of individuals (solution vectors), but of niches, a properly small hyper-rectangle where orthogonal design method likely work well. Each niche selects the best solution found so far as its representative. And orthogonal design method is employed to find potentially good solution which is probably the representative in the niche. The niche mutation, the only genetic operator in this evolutionary algorithm, is guided by the representative of the niche, therefore, the fitness of the offspring is likely better than that of its father, furthermore, ODEA evolves fast. We employ a complex benchmark (moving peaks functions) testing the new approach and the numerical experiments show that ODEA performs much better than SOS [1].